Hyperspaces Of The Regions Below Of Upper Semi-continuous Maps On Non-compact Metric Spaces

During the last decades function spaces have played a prominent role in infinite-dimensional topology. It is clear that the topological structure of the hyperspaces of the regions below of upper semi-continuous maps on compact metric spaces. The aim of this paper is to consider the topological structure of the hyperspaces of the regions below of upper semi-continuous maps on noncompact metric spaces.This paper consists of three chapters.In Chapter 1, we first introduce the developing history of Infinite-Dimensional Topology. And then we list some symbols, conceptions and theorems related to this paper.In Chapter 2, we introduce the study background of this paper and list several important results obtained by some famous scholars. Finally, we list our main results.Let X = (X, d) be a metric space and p be an admissable metric on X×I. Let↓USC(X) denote the family of the regions below of all upper semicontinuous maps from X to I = [0,1] and Cld(X×I) denote the family of all non-empty closed sets in X×I, then↓USC(X) (?) Cld(X×I). For any A, B∈Cld(X×I), we define their Hausdoff distance as follows:We endow Cld(X×I) with the Hausdoff metric topology induced by the Hausdoff distance and call it the hyperspace of the metric space (X×I,ρ).↓USC(X) is topologized as a subspace of the hyperspace Cld(X×I).In Chapter 3, we diccuss some properties of↓USC(X) and show that if X is a connected, complete, non-compact metric space, then↓USC(X) is homeomorphic to the non-separable Hilbert space whose weight is 2^w(X) where w(X) denotes the weight of X; if X is a topologically complete, non-compact metric space and the completion X of X is compact, then↓USC(X) is homeomorphic to the separable Hilbert space l₂.